Question: Factor the following expression: $3$ $x^2+$ $2$ $x$ $-8$
This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(3)}{(-8)} &=& -24 \\ {a} + {b} &=& & & {2} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-24$ and add them together. Remember, since $-24$ is negative, one of the factors must be negative. The factors that add up to ${2}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-4}$ and ${b}$ is ${6}$ $ \begin{eqnarray} {ab} &=& ({-4})({6}) &=& -24 \\ {a} + {b} &=& {-4} + {6} &=& 2 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {3}x^2 {-4}x +{6}x {-8} $ Group the terms so that there is a common factor in each group: $ ({3}x^2 {-4}x) + ({6}x {-8}) $ Factor out the common factors: $ x(3x - 4) + 2(3x - 4) $ Notice how $(3x - 4)$ has become a common factor. Factor this out to find the answer. $(3x - 4)(x + 2)$